Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions

Abstract

We describe a method to derive, from first principles, the long-distance asymptotic behavior of correlation functions of integrable models in the framework of the algebraic Bethe ansatz. We apply this approach to the longitudinal spin–spin correlation function of the XXZ Heisenberg spin- 1/2 chain (with magnetic field) in the disordered regime as well as to the density–density correlation function of the interacting one-dimensional Bose gas. At leading order, the results confirm the Luttinger liquid and conformal field theory predictions.

Long-distance behavior of temperature correlation functions in the one-dimensional Bose gas

“…In the case of the one-dimensional Bose gas at finite temperature, a series of multiple integral representation was obtained in [32] through a scaling limit from the one of the XXZ chain. Here, we work with another one that we obtain from [6] by literarily repeating the steps given in [1] for the case of the XXZ Heisenberg spin chain.…”

“…Moreover, the expansion of this multidimensional contour integral representation into the aforementioned series of multiple integrals does not depend on the specific choice of the state |Ω(Λ) . Hence, even when computing the thermal averages, one can still apply the results of [1]. Therefore,…”

“…The method of analysis we apply builds on the method developed in the work [1]. There, the long-distance asymptotic behavior of the zero temperature correlation functions of massless quantum integrable models was derived in the framework of the algebraic Bethe ansatz.…”

“…In [1], we have focused on the example of the generating function for correlation functions of the third components of local spin in the XXZ spin-1/2 Heisenberg chain. We have mentioned, however, that the method can be applied to other integrable models as well.…”

Recursive formulas for the overlaps between Bethe states and product states in XXZ Heisenberg chains

“…Among the most remarkable results obtained by means of the Algebraic Bethe Ansatz, we must mention the proof of the Gaudin formula for the norm of a Bethe state [68], the Slavnov formula [69] which is a a general expression for the scalar product between two Bethe states of the form (17) in which only one of the two sets of rapidities satisfies the Bethe equations, and the determination of the form factors of the most relevant operators [48,49,51].…”